SEERAVERSE RH Program: Closing the Finite Gap: From All-Depth Pick Positivity to Analytic Rigidity

This research note records a major internal milestone of the SEERAVERSE RH PROGRAM. The program studies a route to the Riemann Hypothesis in which the problem is reduced to positivity of a family of finite boundary Pick matrices. The former finite-middle obstruction has now been discharged analytically by a variable reflection-cap argument. An effective centrality estimate yields a stage-dependent cap on the fresh reflection coefficient; the associated propagation factor admits an exact factorization with a telescoping geometric term. Combined with the threshold contraction \(T_{n+1}/T_n<4/9\) and a proof-grade base margin, this gives a conservative uniform margin exceeding \(10^{507}\) throughout \(4053\le n\le10148\). Together with the proof-grade prefix through \(n=4052\) and an effective restart for \(n\ge10149\), this yields all-depth finite Pick positivity internally. The downstream analytic chain proceeds through positive-real interpolation, Montel compactness, Nevanlinna-class uniqueness on a non-Blaschke lattice, meromorphic identity continuation, and pole exclusion. The complete dependency chain has passed an internal line-by-line audit. No public claim that the Riemann Hypothesis has been proved is made in this note; independent external mathematical verification is requested.